20. a snack machine sells granola bars and bags of trail mix. the machine holds, at most, 200 units of…

20. a snack machine sells granola bars and bags of trail mix. the machine holds, at most, 200 units of snacks. at least 4 granola bars are sold for each bag of trail mix. each granola bar sells for $1.00 and each bag of trail mix sells for $1.25. determine the amount of granola bars and trail mix bars required to sell to maximize revenue.

20. a snack machine sells granola bars and bags of trail mix. the machine holds, at most, 200 units of snacks. at least 4 granola bars are sold for each bag of trail mix. each granola bar sells for $1.00 and each bag of trail mix sells for $1.25. determine the amount of granola bars and trail mix bars required to sell to maximize revenue.

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of granola - bars and $y$ be the number of bags of trail - mix.

Step2: Set up constraints

We have the following constraints:

  1. Capacity constraint: $x + y\leq200$.
  2. Minimum granola - bar constraint: $x\geq4$. Also, $x\geq0,y\geq0$ since the number of items cannot be negative.

Step3: Set up the revenue function

The revenue function $R$ is given by $R = 1x+1.25y$.

Step4: Rewrite the capacity constraint for $y$

$y\leq200 - x$.

Step5: Analyze the revenue function

Since the revenue function $R=x + 1.25y$, and $y$ has a higher coefficient than $x$, we want to sell as many bags of trail - mix as possible while still satisfying the constraints. From the capacity constraint $x + y=200$ (to maximize the total number of items sold) and considering $x\geq4$. We set $x = 4$.

Step6: Calculate the value of $y$

If $x = 4$, then from $x + y=200$, we substitute $x$ into the equation: $4+y=200$, so $y=196$.

Step7: Calculate the revenue

$R=1\times4 + 1.25\times196=4+245=249$.

Answer:

4 granola bars and 196 bags of trail - mix should be sold to maximize revenue.